| L(s) = 1 | + 1.70·2-s − 3.27·3-s + 0.900·4-s + 0.246·5-s − 5.57·6-s − 1.87·8-s + 7.73·9-s + 0.420·10-s + 11-s − 2.94·12-s + 3.17·13-s − 0.808·15-s − 4.99·16-s + 6.49·17-s + 13.1·18-s + 4.32·19-s + 0.222·20-s + 1.70·22-s + 3.15·23-s + 6.13·24-s − 4.93·25-s + 5.39·26-s − 15.5·27-s + 6.48·29-s − 1.37·30-s − 1.78·31-s − 4.75·32-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 1.89·3-s + 0.450·4-s + 0.110·5-s − 2.27·6-s − 0.662·8-s + 2.57·9-s + 0.132·10-s + 0.301·11-s − 0.851·12-s + 0.879·13-s − 0.208·15-s − 1.24·16-s + 1.57·17-s + 3.10·18-s + 0.991·19-s + 0.0496·20-s + 0.363·22-s + 0.658·23-s + 1.25·24-s − 0.987·25-s + 1.05·26-s − 2.98·27-s + 1.20·29-s − 0.251·30-s − 0.319·31-s − 0.839·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477983759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477983759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 0.246T + 5T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 1.78T + 31T^{2} \) |
| 37 | \( 1 - 8.38T + 37T^{2} \) |
| 41 | \( 1 - 0.553T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 6.45T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 + 3.65T + 67T^{2} \) |
| 71 | \( 1 - 0.345T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 + 3.77T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 - 0.246T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.32585597497000310578345675861, −10.19104368237721659494061888671, −9.417672519791143516620521807235, −7.76766937851678235964188845323, −6.59686871803777122421118963724, −5.92351834234290719270955532607, −5.32323241809895874119978885971, −4.44741285580462308283753419215, −3.39527825713004395826329299551, −1.08292731508802072619681566498,
1.08292731508802072619681566498, 3.39527825713004395826329299551, 4.44741285580462308283753419215, 5.32323241809895874119978885971, 5.92351834234290719270955532607, 6.59686871803777122421118963724, 7.76766937851678235964188845323, 9.417672519791143516620521807235, 10.19104368237721659494061888671, 11.32585597497000310578345675861
